Can I take q10 if I take thyroxine?

Yes. Coenzyme q10 is presently being peddled for pretty much everything and if you believed what these people are saying you would at best be reading uncritically. Coenzyme q specifically helps a group of very rare diseases. Supplement makers are not held by law to the standards of truth that the healthtap team is. If you want to pay money to people for deceiving you, that's your business.
Yes, but why? Yes you can, but it won't do much. Why are you planning on taking it in the first place? What are you attempting to treat? How are you going to tell if it's better. Many of the supplements have almost no active ingredient in them. Don't get your medical information from tv and the internet.